Distributional reinforcement learning improves performance by effectively capturing environmental stochasticity, but a comprehensive theoretical understanding of its effectiveness remains elusive. In this paper, we present a regret analysis for distributional reinforcement learning with general value function approximation in a finite episodic Markov decision process setting. We first introduce a key notion of Bellman unbiasedness for a tractable and exactly learnable update via statistical functional dynamic programming. Our theoretical results show that approximating the infinite-dimensional return distribution with a finite number of moment functionals is the only method to learn the statistical information unbiasedly, including nonlinear statistical functionals. Second, we propose a provably efficient algorithm, $\texttt{SF-LSVI}$, achieving a regret bound of $\tilde{O}(d_E H^{\frac{3}{2}}\sqrt{K})$ where $H$ is the horizon, $K$ is the number of episodes, and $d_E$ is the eluder dimension of a function class.

eBay partners with external vendors, which allows customers to freely select a vendor to complete their eBay experiences. However, vendor outages can hinder customer experiences. Consequently, eBay can disable a problematic vendor to prevent customer loss. Disabling the vendor too late risks losing customers willing to switch to other vendors, while disabling it too early risks losing those unwilling to switch. In this paper, we propose a data-driven solution to answer whether eBay should disable a problematic vendor and when to disable it. Our solution involves forecasting customer behavior. First, we use a multiplicative seasonality model to represent behavior if all vendors are fully functioning. Next, we use a Monte Carlo simulation to represent behavior if the problematic vendor remains enabled. Finally, we use a linear model to represent behavior if the vendor is disabled. By comparing these forecasts, we determine the optimal time for eBay to disable the problematic vendor.

Factors which attract customers and persuade them to buy new car are various regarding different consumer tastes. There are some methods to extract pattern form mass data. In this case we firstly asked passenger car marketing experts to rank more important factors which affect customer decision making behavior using fuzzy Delphi technique, then we provided a sample set from questionnaires and tried to apply a useful artificial neural network method called selforganizing map (SOM) to find out which factors have more effect on Iranian customer's buying decision making. Fuzzy tools were applied to adjust the study to be more real. MATLAB software was used for developing and training network. Results report four factors are more important rather than the others. Results are rather different from marketing expert rankings. Such results would help manufacturers to focus on more important factors and increase company sales level.

In order to sample from an unnormalized probability density function, we propose to combine continuous normalizing flows (CNFs) with rejection-resampling steps based on importance weights. We relate the iterative training of CNFs with regularized velocity fields to a JKO scheme and prove convergence of the involved velocity fields to the velocity field of the Wasserstein gradient flow (WGF). The alternation of local flow steps and non-local rejection-resampling steps allows to overcome local minima or slow convergence of the WGF for multimodal distributions. Since the proposal of the rejection step is generated by the model itself, they do not suffer from common drawbacks of classical rejection schemes. The arising model can be trained iteratively, reduces the reverse Kulback-Leibler (KL) loss function in each step, allows to generate iid samples and moreover allows for evaluations of the generated underlying density. Numerical examples show that our method yields accurate results on various test distributions including high-dimensional multimodal targets and outperforms the state of the art in almost all cases significantly.

Bayesian Inference with Markov Chain Monte Carlo requires efficient computation of the likelihood function. In some scientific applications, the likelihood must be computed by numerically solving a partial differential equation, which can be prohibitively expensive. We demonstrate that some such problems can be made tractable by amortizing the computation with a surrogate likelihood function implemented by a neural network. We show that this has two additional benefits: reducing noise in the likelihood evaluations and providing fast gradient calculations. In experiments, the approach is applied to a model of heliospheric transport of galactic cosmic rays, where it enables efficient sampling from the posterior of latent parameters in the Parker equation.

This paper proposes to establish the distance between partial preference orderings based on two very different approaches. The first approach corresponds to the brute force method based on combinatorics. It generates all possible complete preference orderings compatible with the partial preference orderings and calculates the Frobenius distance between all fully compatible preference orderings. Unfortunately, this first method is not very efficient in solving high-dimensional problems because of its big combinatorial complexity. That is why we propose to circumvent this problem by using a second approach based on belief functions, which can adequately model the missing information of partial preference orderings. This second approach to the calculation of distance does not suffer from combinatorial complexity limitation. We show through simple examples how these two theoretical methods work.

We consider the problem of Bayesian regression with trustworthy uncertainty quantification. We define that the uncertainty quantification is trustworthy if the ground truth can be captured by intervals dependent on the predictive distributions with a pre-specified probability. Furthermore, we propose, Trust-Bayes, a novel optimization framework for Bayesian meta learning which is cognizant of trustworthy uncertainty quantification without explicit assumptions on the prior model/distribution of the functions. We characterize the lower bounds of the probabilities of the ground truth being captured by the specified intervals and analyze the sample complexity with respect to the feasible probability for trustworthy uncertainty quantification. Monte Carlo simulation of a case study using Gaussian process regression is conducted for verification and comparison with the Meta-prior algorithm.

This paper describes a discrete state-space model -- and accompanying methods -- for generative modelling. This model generalises partially observed Markov decision processes to include paths as latent variables, rendering it suitable for active inference and learning in a dynamic setting. Specifically, we consider deep or hierarchical forms using the renormalisation group. The ensuing renormalising generative models (RGM) can be regarded as discrete homologues of deep convolutional neural networks or continuous state-space models in generalised coordinates of motion. By construction, these scale-invariant models can be used to learn compositionality over space and time, furnishing models of paths or orbits; i.e., events of increasing temporal depth and itinerancy. This technical note illustrates the automatic discovery, learning and deployment of RGMs using a series of applications. We start with image classification and then consider the compression and generation of movies and music. Finally, we apply the same variational principles to the learning of Atari-like games.

Models for categorical sequences typically assume exchangeable or first-order dependent sequence elements. These are common assumptions, for example, in models of computer malware traces and protein sequences. Although such simplifying assumptions lead to computational tractability, these models fail to capture long-range, complex dependence structures that may be harnessed for greater predictive power. To this end, a Bayesian modelling framework is proposed to parsimoniously capture rich dependence structures in categorical sequences, with memory efficiency suitable for real-time processing of data streams. Parsimonious Bayesian context trees are introduced as a form of variable-order Markov model with conjugate prior distributions. The novel framework requires fewer parameters than fixed-order Markov models by dropping redundant dependencies and clustering sequential contexts. Approximate inference on the context tree structure is performed via a computationally efficient model-based agglomerative clustering procedure. The proposed framework is tested on synthetic and real-world data examples, and it outperforms existing sequence models when fitted to real protein sequences and honeypot computer terminal sessions.

Investigating noise distribution beyond Gaussian in diffusion generative models is an open problem. The Gaussian case has seen success experimentally and theoretically, fitting a unified SDE framework for score-based and denoising formulations. Recent studies suggest heavy-tailed noise distributions can address mode collapse and manage datasets with class imbalance, heavy tails, or outliers. Yoon et al. (NeurIPS 2023) introduced the L\'evy-Ito model (LIM), extending the SDE framework to heavy-tailed SDEs with $\alpha$-stable noise. Despite its theoretical elegance and performance gains, LIM's complex mathematics may limit its accessibility and broader adoption. This study takes a simpler approach by extending the denoising diffusion probabilistic model (DDPM) with $\alpha$-stable noise, creating the denoising L\'evy probabilistic model (DLPM). Using elementary proof techniques, we show DLPM reduces to running vanilla DDPM with minimal changes, allowing the use of existing implementations with minimal changes. DLPM and LIM have different training algorithms and, unlike the Gaussian case, they admit different backward processes and sampling algorithms. Our experiments demonstrate that DLPM achieves better coverage of data distribution tail, improved generation of unbalanced datasets, and faster computation times with fewer backward steps.